Statistical Meaning of Mean Functions

The basic properties of the Fisher information allow to reveal the statistical meaning of classical inequalities between mean functions. The properties applied to scale mixtures of Gaussian distributions lead to a new mean function of purely statistical origin, unrelated to the classical arithmetic, geometric, and harmonic means. We call it the informational mean and show that when the arguments of the mean functions are Hermitian positive definite matrices, not necessarily commuting, the informational mean lies between the arithmetic and harmonic means, playing, in a sense, the role of the geometric mean that cannot be correctly defined in case of non-commuting matrices.\\ Surprisingly the monotonicity and additivity properties of the Fisher information lead to a new generalization of the classical inequality between the arithmetic and harmonic means.